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<article class="li"><h3 class="heading">
<span class="type">Item</span><span class="space"> </span><span class="codenumber">1</span><span class="period">.</span>
</h3>
<p>If <span class="process-math">\(\lambda&lt;0\text{,}\)</span> we write <span class="process-math">\(\lambda=-k^2\text{,}\)</span> where <span class="process-math">\(k=\sqrt{|\lambda|}&gt;0\text{,}\)</span>Then the characteristic equation yields <span class="process-math">\(r^2 =k^2~\to~r=\pm k\)</span> and the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=c_1e^{kx} + c_2e^{-kx}.
\end{equation*}
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<p class="continuation">By boundary conditions, we must have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
c_1 + c_2 = 0,~~ c_1e^{kL} + c_2e^{-kL} = 0,
\end{equation*}
</div>
<p class="continuation">which gives the solution <span class="process-math">\(c_1 = c_2 = 0\text{.}\)</span> Then <span class="process-math">\(y(x) = 0\text{,}\)</span> which is a trivial solution. Discard it.</p></article><span class="incontext"><a href="sec7_2.html#li-27" class="internal">in-context</a></span>
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